In this paper we determine the computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(logn) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangen...

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph...

I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application, I determine which rational double points do appear on geometric generic fibers.

We show that if the image of a Legendrian submanifold under contact homeomorphism (i.e. is $C^0$-limit contactomorphisms) smooth then it Legendrian, assuming only positive local lower bounds on conformal factors approximating contactomorphisms. More generally analogous result holds for coisotropic submanifolds in sense arXiv:1306.6367. This version Humili\`ere-Leclercq-Seyfaddini rigidity theor...

We outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a standard basis.

Because of the hairy ball theorem, the only closed 2-manifold that supports a lattice in its tangent space is T . But, if singular points (i.e. points whose tangent space is not endowed with 2 distinct coordinate directions) are allowed, then it becomes possible to give the tangent space a lattice. Because the lattice is well defined everywhere around the points, the effect of moving around the...